NEW YORK (Reuters) - Oil prices fell about 2 percent in extended trading on Tuesday as data showed a surprise build in U.S. crude inventories last week, adding to the market's nervousness around Britain's vote next week on whether to leave the European Union.

Data from the American Petroleum Institute (API) showed U.S. crude inventories rose by 1.2 million barrels in the week to June 10 to 536.7 million, compared with analysts' expectations for a decrease of 2.3 million barrels. [API/S]

"It (the build) goes along with all the uncertainty in the market right now ... nothing is solid," Carl Larry, director of business development for oil and gas at Frost & Sullivan, said.

Brent <LCOc1> and U.S. crude <CLc1> deepened losses in post-settlement trading, both falling by more than $1 a barrel each.

Prices ended the session lower for the fourth straight day with Brent down 52 cents at $49.83 a barrel, while U.S. crude closed 39 cents lower at $48.49. Gasoline futures <RBc1> touched a more than one-month low of $1.50 a gallon during the session.

U.S. crude traded between $48.69 and $48.02 during the session, in the tightest range in nearly two years, traders said.

Safe-haven German Bund yields <DE10YT=RR> fell below zero for the first time, while industrial commodities and equity markets, seen as more vulnerable to economic risk, dropped after polls showed Britain's "Leave" campaign leading ahead of the June 23 vote on EU membership.

The referendum-related concerns eclipsed an upbeat forecast for oil demand growth from the International Energy Agency (IEA), which said the oil market is essentially balanced after two years of surpluses. [IEA/M] On Monday, OPEC forecast that the oil market would be more balanced in the second half of 2016 as outages in Nigeria and Canada help to speed erosion of a supply glut.

The campaign for Britain to leave the EU has a "significant lead" ahead of the referendum, and about 47 percent of likely voters said they will opt to leave, compared with 40 percent who want to stay, according to a poll of 2,497 people.

"This (the referendum) has rattled a lot of financial and commodity trader/investors with money seemingly starting to flow to the so-called safe-haven U.S. dollar until the dust settles and the voting is concluded," said Dominick Chirichella, senior partner at the Energy Management Institute in New York.

"The strong U.S. dollar versus most currency pairs is a negative price directional driver for the oil complex."

If Britain voted to leave the EU, a prospect dubbed "Brexit", investors fear the bloc could slip into recession, which in turn could undermine oil demand.

The dollar <.DXY> has risen about 1.7 percent from its June lows against a basket of currencies, spurred by Brexit fears.

Adding to the uncertainty in the markets was the ongoing two-day U.S. central bank meeting.

"Not increasing interest rates would be interpreted as the U.S. economy is not healthy enough for increasing interest rates," a U.S. trader said.

(Additional reporting by Amanda Cooper in London, Henning Gloystein in Singapore and Aaron Sheldrick in Tokyo; Editing by Tom Brown and Andrew Hay)